This chapter tells an interesting story on how playing with a bicycle wheel can connect to a fundamental theorem from differential geometry. The internal angles in a planar triangle add up to 180°. This fact can be restated in a more general and yet more basic way: if I walk around a closed curve in the plane, then my nose, treated as a vector, will rotate by 2*π*(provided that I always look straight ahead).¹

Does the same hold for an inhabitant of a curved surface? Figure 10.1 shows a triangular path on the sphere. Two of the sides lie

EP - 147 PB - Princeton University Press PY - 2009 SN - 9780691154565 SP - 133 T2 - The Mathematical Mechanic T3 - Using Physical Reasoning to Solve Problems UR - http://www.jstor.org/stable/j.ctt7rjgk.12 Y2 - 2021/06/21/ ER -